Problem 75 - Prealgebra

Question

Apply the distributive property, then find the LCD and simplify. $$\frac{2 x}{5}+\frac{5 x}{8}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{41x}{40} \).

Step 1: Identify the Terms

We begin with the expression \( \frac{2x}{5} + \frac{5x}{8} \). These are two fractions with different denominators that we need to combine.

Step 2: Find the Least Common Denominator (LCD)

The denominators of the fractions are 5 and 8. To find the least common denominator, find the least common multiple of these two numbers, which is 40.

Step 3: Rewrite Each Fraction with the LCD

Convert each fraction to have a denominator of 40:\[ \frac{2x}{5} = \frac{2x \times 8}{5 \times 8} = \frac{16x}{40} \]\[ \frac{5x}{8} = \frac{5x \times 5}{8 \times 5} = \frac{25x}{40} \]

Step 4: Add the Fractions

Since both fractions now have the same denominator, we can add them together:\[ \frac{16x}{40} + \frac{25x}{40} = \frac{16x + 25x}{40} = \frac{41x}{40} \]

Step 5: Simplify If Necessary

The fraction \( \frac{41x}{40} \) is already in its simplest form since 41 and 40 have no common factors other than 1.

Key Concepts

Understanding the Least Common Denominator

In the world of fractions, the least common denominator (LCD) is a crucial concept. It is the smallest number that is a multiple of the denominators of the two fractions involved.

Think of the LCD as a common platform where both fractions can 'meet' and be compared or combined. Here’s how you find it:

• List the multiples of each denominator.
• Identify the smallest multiple that both denominators share.

For example, if we have the denominators 5 and 8, we list their multiples:

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
• Multiples of 8: 8, 16, 24, 32, 40...

As you can see, 40 is the first common multiple they both hit. Hence, 40 is the least common denominator that we will use to rewrite each fraction in the same format.

The Art of Fraction Addition

When adding fractions, they must have the same denominator to combine them directly. If they have different denominators:

• Convert each fraction to an equivalent fraction with the LCD as the new denominator.
• Adjust the numerators accordingly by multiplying them by the factor that will change their old denominator into the LCD.

For example, to add \(\frac{2x}{5}\) and \(\frac{5x}{8}\):

• First, change \(\frac{2x}{5}\) to \(\frac{16x}{40}\) by multiplying both the numerator and the denominator by 8.
• Then, convert \(\frac{5x}{8}\) to \(\frac{25x}{40}\) by multiplying both the numerator and the denominator by 5.
• Now, you can add \(\frac{16x}{40}\) and \(\frac{25x}{40}\) to get \(\frac{41x}{40}\).

This process allows for a straightforward addition of fractions once they share a common base or denominator. The numerators simply add together, while the denominator remains unchanged in the result.

Mastering Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its smallest form, ensuring that the numerator and the denominator have no common factors other than 1. This process makes the fraction easier to read and work with.

To simplify, you:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.

However, in our exercise, the resultant fraction \(\frac{41x}{40}\) is already in its simplest form. Why? Because the numbers 41 and 40 have no common factors, other than 1. This makes our job easy—when you can't break down the numbers any further, you've reached the simplest form.

• Ensures clarity in mathematical expressions.
• Prevents unnecessary complexity.

This principle is key in making fractions manageable and tidy in mathematical operations.